Scottish Higher Mathematics

Higher Mathematics

Revise the current Scottish Higher Mathematics course with clear explanations, original worked examples and interactive questions. The topic library covers algebra, functions, coordinate geometry, trigonometry, calculus, vectors and recurrence relations, with printable and teacher-ready support where available.

Topic practice Higher Maths Mixed Revision Formula reference

Topic library

Topic pages use Learn, Practise, Check and Answers sections with Higher Mathematics notation, short worked examples, generated practice and mixed revision for building exam-ready methods.

Higher Maths Revision

Use focused topic questions first, then choose balanced or calculus-focused mixed revision to practise method selection across the course.

Topic practice Mixed revision Readiness check

Higher Maths Formulae

Keep the formula reference open while revising rules, exact notation and the links between algebra, trigonometry and calculus.

Open formula reference

Higher Maths Worksheets

Teachers can build printable sets from the same checked Higher question families used in pupil practice.

Higher Maths for Teachers

Set homework, use diagnostic starters and connect pupil follow-up to the matching public revision route.

Algebra and Functions

Functions, graph transformations, Polynomials and Quadratics, intersections, Exponentials and Logarithms, and mathematical models.

Open practice

Function notation and evaluation

Use f(x), substitute values, interpret outputs and connect functions to graphs.

Substitute accurately and state outputs clearly.

Composite and inverse functions

Build composite functions, find inverses and state suitable domains and ranges.

Work in the correct order and check the inverse.

Domain, range and transformations

State domains and ranges, then sketch translations, reflections and stretches of functions.

Link function notation to graph movement.

Quadratic equations and discriminant

Solve quadratic equations and use the discriminant to determine or control the nature of roots.

Use b² - 4ac and interpret its sign.

Completing the square and quadratic inequalities

Complete the square for unitary and non-unitary quadratics, then solve quadratic inequalities.

Expose graph features and use a sign diagram.

Factor and remainder theorem

Evaluate f(a), identify linear factors and find remainders of polynomial division.

Test roots by substitution.

Cubic and quartic equations

Factorise and solve cubic or quartic polynomial equations using known or discoverable factors.

Find one factor, divide, then solve the reduced equation.

Intersections of functions

Find the coordinates where a line and curve, or two curves, have equal outputs.

Set the functions equal and find every coordinate.

Exponential and logarithmic functions

Connect exponential and logarithmic functions as inverses and interpret their graphs.

Switch between index and logarithmic form.

Log laws and equations

Simplify logarithmic expressions and solve exponential or logarithmic equations with valid domains.

Use one law at a time and check arguments.

Exponential modelling and linearisation

Model growth and decay, find constants from data and linearise power or exponential relationships using logarithms.

Match a straight-line form to transformed variables.

Sketching polynomials

Use roots, intercepts, stationary points and end behaviour to sketch polynomial graphs.

Mark roots and end behaviour first.

Coordinate Geometry and Circles

Straight Line and Circles methods: coordinate formulae, geometrical line problems, circle equations, tangents and intersections.

Open practice

Straight line methods

Find gradients and equations, including parallel and perpendicular lines and m = tan θ.

Choose a point-gradient or gradient-intercept form.

Distance and midpoint

Calculate distances and midpoints accurately and use them inside multi-step geometry problems.

Use coordinate differences in the correct order.

Geometrical line problems

Use medians, altitudes, perpendicular bisectors and line intersections in coordinate geometry.

Translate the geometric property into a gradient and point.

Equation of a circle

Find a circle's centre and radius from standard or expanded form and build its equation.

Complete the square in both coordinates.

Tangents to circles

Use the perpendicular radius and tangent property to form tangent equations.

Find the radius gradient before the tangent gradient.

Line and circle intersections

Find and interpret line-circle and circle-circle intersections, including tangency cases.

Substitute or subtract equations, then check every point.

Trigonometry

Exact values, identities, equations in degrees and radians, formulae, graphs and wave functions.

Open practice

Exact values and identities

Use exact trigonometric values and the identity sin² x + cos² x = 1.

Keep exact values until a decimal is requested.

Solving trigonometric equations

Solve trigonometric equations in degrees or radians over a stated interval.

Find every solution in the interval.

Addition and double-angle formulae

Apply addition formulae, double-angle formulae and related identities to simplify or solve.

Choose the formula that matches the expression.

Wave functions

Convert a cos x + b sin x to a single sine or cosine form and use it to find ranges and solve equations.

Use k² = a² + b² and compare coefficients.

Trigonometric graphs

Interpret amplitude, period, phase shift and related transformations of sine, cosine and tangent.

Read horizontal transformations carefully.

Differentiation

Power and trigonometric derivatives, chain rule, tangents, rates, stationary points, optimisation and graph behaviour.

Open practice

Differentiating powers

Differentiate polynomial and algebraic terms written as powers of x.

Multiply by the power and reduce it by 1.

Trigonometric differentiation

Differentiate sin(kx) and cos(kx), including the sign and inner multiplier

Differentiate in radians and include the inner factor.

Chain rule

Differentiate composite algebraic and trigonometric functions using the chain rule.

Differentiate the outer function, then multiply by the inner derivative.

Tangents, normals and rates of change

Find tangent and normal equations and interpret derivatives as rates of change.

Differentiate, substitute, then use straight-line methods.

Stationary points and optimisation

Find and classify stationary points and solve contextual optimisation problems.

Set dy/dx = 0 and interpret the result.

Increasing, decreasing and derivative graphs

Use the sign and graph of a derivative to describe where a function rises, falls or is stationary.

Read f'(x) above, on or below the axis.

Maximum and minimum on closed intervals

Compare stationary values with endpoint values to find greatest and least values on a closed interval.

Test every stationary point and both endpoints.

Integration

Polynomial, shifted-power and trigonometric integration, definite integrals, areas and recovering functions from rates.

Open practice

Indefinite integration

Reverse the power rule for algebraic expressions and include the constant of integration.

Increase the power and divide by the new power.

Integration of shifted powers

Integrate (x + q)ⁿ and (px + q)ⁿ with the correct inner-factor adjustment.

Divide by p(n + 1).

Trigonometric integration

Integrate p sin(qx + r) and p cos(qx + r), keeping the sign and inner factor correct

Divide by q and check by differentiating.

Definite integrals and area

Evaluate definite integrals and find area between a curve and the x-axis.

Use F(b) - F(a) and check the sign.

Area between curves

Find intersections and integrate top function minus bottom function between the correct limits.

Sketch or compare functions before integrating.

Reverse derivatives with conditions

Recover a function from its derivative and use an initial condition to find the constant.

Integrate first, then apply the condition.

Differential equations and rates

Solve simple equations of the form dy/dx = f(x) and interpret initial-condition models.

Integrate the rate and use the given state.

Vectors

Two- and three-dimensional vectors, magnitude, unit vectors, scalar product and geometric proofs.

Open practice

Vector operations and magnitude

Add, subtract and scale 2D or 3D vectors, then calculate their magnitudes.

Work component by component.

3D and unit vectors

Use i, j, k notation and find unit vectors in two or three dimensions.

Divide a non-zero vector by its magnitude.

Position vectors and vector geometry

Use position vectors and vector pathways to solve geometric problems.

State each vector route clearly.

Collinearity and internal division

Prove collinearity and find points that divide a line internally in a given ratio.

Show a common scalar multiple or use the section formula.

Scalar product

Calculate scalar products, test perpendicularity and find the angle between vectors.

Use a · b = |a||b| cos θ.

Recurrence Relations

Recurrence Relations and Sequences: generate terms, derive recurrence rules from contexts and interpret limiting behaviour.

Open practice

Recurrence relations

Calculate successive terms accurately from a starting value and recurrence rule.

Use the previous term each time.

Recurrence modelling and limits

Derive linear recurrence models, find limits where they exist and interpret them in context.

Model the update, then solve L = aL + b.

Sequences and limits

Review sequence notation and use successive values to support recurrence-limit reasoning.

Describe the long-term behaviour precisely.

Mixed Revision and Problem Solving

Applications and Problem Solving across Higher Mathematics, requiring method choice, interpretation and clear reasoning.

Open practice

Mixed exam-style problem solving

Combine algebra, geometry, calculus, trigonometry, vectors and recurrence in original multi-step problems.

Choose the method before calculating.